<PRE><B>Question 195816</B>


Looking at the expression {{{x^2-9x+8}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-9}}}, and the last term is {{{8}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{8}}} to get {{{(1)(8)=8}}}.



Now the question is: what two whole numbers multiply to {{{8}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-9}}}?



To find these two numbers, we need to list &lt;font size=4&gt;&lt;b&gt;all&lt;/b&gt;&lt;/font&gt; of the factors of {{{8}}} (the previous product).



Factors of {{{8}}}:

1,2,4,8

-1,-2,-4,-8



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{8}}}.

1*8
2*4
(-1)*(-8)
(-2)*(-4)


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-9}}}:



&lt;table border=&quot;1&quot;&gt;&lt;th&gt;First Number&lt;/th&gt;&lt;th&gt;Second Number&lt;/th&gt;&lt;th&gt;Sum&lt;/th&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+8=9&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+4=6&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-1+(-8)=-9&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+(-4)=-6&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-1}}} and {{{-8}}} add to {{{-9}}} (the middle coefficient).



So the two numbers {{{-1}}} and {{{-8}}} both multiply to {{{8}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-9}}}



Now replace the middle term {{{-9x}}} with {{{-x-8x}}}. Remember, {{{-1}}} and {{{-8}}} add to {{{-9}}}. So this shows us that {{{-x-8x=-9x}}}.



{{{x^2+highlight(-x-8x)+8}}} Replace the second term {{{-9x}}} with {{{-x-8x}}}.



{{{(x^2-x)+(-8x+8)}}} Group the terms into two pairs.



{{{x(x-1)+(-8x+8)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x-1)-8(x-1)}}} Factor out {{{8}}} from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



{{{(x-8)(x-1)}}} Combine like terms. Or factor out the common term {{{x-1}}}


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Answer:



So {{{x^2-9x+8}}} factors to {{{(x-8)(x-1)}}}.



Note: you can check the answer by FOILing {{{(x-8)(x-1)}}} to get {{{x^2-9x+8}}} or by graphing the original expression and the answer (the two graphs should be identical).
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